Compensation of polarization mode dispersion in single mode fiber for maximum-likelihood sequence estimation

ABSTRACT

An output signal of a single mode fiber (SMF) is spectrally shaped to achieve characteristics of a predefined channel “target” response. The target response is that of a partial-response, maximum-likelihood channel with additive white Gaussian noise. A receiver employs a maximum-likelihood sequence estimation (MLSE) detector having its detection algorithm, such as a Viterbi algorithm (VA), matched to the target response. Thus, state, branch, and path metric calculations for a Viterbi trellis may be optimized for a channel having this target response.

CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application claims the benefit of the filing date of U.S.provisional application No. 60/280,326, filed on Mar. 30, 2001 asattorney docket no. Bessios 3PROV.

BACKGROUND OF THE INVENTION

[0002] 1. Field of the Invention

[0003] The present invention relates to detection of data in acommunications system, and, more particularly, to compensation forsignal dispersion in an optical fiber.

[0004] 2. Description of the Related Art

[0005] In many digital communications systems, a user generates digitalinformation that is then processed into an encoded (e.g.,error-correction encoded) and/or packetized stream of data. The streamof data is then divided into discrete blocks. Each of the blocks may bemapped onto a corresponding one of a sequence of code or symbol values(“symbols”) chosen from a pre-defined alphabet A, and generated with aperiod T_(s), sometimes referred to as the “baud” rate. For opticaltransmission of the digital information, an optical carrier operatingwith a wavelength of, for example, 1310 nm or 1550 nm, is modulated withthe encoded stream of data. The modulated optical carrier is then passedthrough an optical fiber, such as a single mode fiber (SMF) having itslowest order bound propagating at 1310 nm or 1550 nm.

[0006] The modulated optical signal transmitted through the opticalfiber channel comprises a series of light pulses. Since a transmissionmedium may be modeled as a filter having a corresponding(frequency-domain) transfer function and (time-domain) impulse response,the pulse transmitted through the channel may have its shape modifiedbased on this transfer function. The analog pulse shape may be modifiedin amplitude and phase, and also experience dispersion of the pulse.Consequently, the time duration of the pulse transmitted through amedium may extend over a period of time greater than the time durationof a particular symbol. Adjacent pulses transmitted through the mediummay thus corrupt each other, which corruption is known as inter-symbolinterference (ISI). This characteristic of the optical fiber (channel)is termed memory (e.g., if one adjacent pulse contributes to ISI, thememory “length” is one).

[0007] As bit rates in optical communication systems increase for highspeed data transmission, such as rates above 10 Gbps, receivers areincreasingly faced with mitigation of effects of pulse dispersion andISI to the optical signal passing through the optical fiber channel. Areceiver typically includes a detector forming decisions for received,sampled channel output data (“receive signal”) corresponding to thetransmitted pulses. These detectors may apply compensation/equalizationto the receive signal, and employ algorithms such as maximum-likelihoodsequence detection to reconstruct the sequence of pulses in thetransmitted, encoded stream of data.

[0008] For optical communication systems, there are several sources ofpulse dispersion through an SMF. One source of dispersion is chromaticdispersion that causes time-domain pulse broadening due to the differenttraveling velocities of each of the optical pulse's spectral components.Prior art methods of compensation for chromatic dispersion use anopposite dispersion-compensating fiber (DCF) that has a greaterdispersion parameter, usually by a factor of 10.

[0009] Another source of dispersion is polarization mode dispersion(PMD) that arises from imperfections in the circular symmetry of thefiber core. The imperfections typically are caused by manufacturingimperfections in the core, chemical impurities, and excessive bending orstrain during installation. Imperfect circular symmetry results inbirefringent SMF that causes two orthogonal principal polarizationstates (PPS) to propagate with different velocity through the fibercore. The resulting average differential group delay (DGD) isproportional to the square-root of the transmission distance. Forexample, an SMF having PMD of 10 ps/{square root}{square root over (L)},where L is the distance in kilometers, has a DGD of 100 ps at a distanceof 100 km. A distances greater than 100 km and bit rates of greater than10 Gbps, the DGD becomes significant when compared to the symbol period(T_(S)=100 ps at 10 Gbps). For a 10 Gbps transmission data rate, themagnitude of the maximum allowable value for the mean DGD (τ_(d))between two orthogonal PPSs has an upper bound of 100 ps (i.e.,(τ_(d))≦100 ps). The (SMF) channel may be modeled as a two-symboldispersive channel with impulse response h(t), and the model for a 10Gbps channel that reflects statistical differential delay between thetwo PPS components may be as given in equation (1):

h(t)={square root}{square root over (α)}(δ(t−τ _(d)))+{squareroot}{square root over (1−α)}(δ(t))  (1)

[0010] δ(·) defined as the delta function, (α/1−α)) defined as the powerdistribution ratio among the two orthogonal PPSs with 0<α<1, and whereτ_(d) follows a Maxwell distribution as given in equation (2):$\begin{matrix}\begin{matrix}{{P_{\langle\tau_{d}\rangle}\left( \tau_{d} \right)} = {\frac{32\tau_{d}^{2}}{\pi^{2}{\langle\tau_{d}\rangle}^{3}}{\exp \left( {- \frac{4\tau_{d}^{2}}{\pi {\langle\tau_{d}\rangle}^{2}}} \right)}}} & {0 < \tau_{d} < \infty}\end{matrix} & (2)\end{matrix}$

[0011] ISI results in multi-level channel output values due to theeffect of the channel's memory on the binary input levels, causingdegradation of a receiver's bit error rate (BER) performance. Linear ISIand time-varying PMD effects are generally compensated for usingadaptive equalization.

SUMMARY OF THE INVENTION

[0012] The present invention relates to spectrally shaping an outputsignal of a single mode fiber (SMF) to achieve characteristics of apredefined channel “target” response. The target response is that of apartial-response channel with additive white Gaussian noise. A receivermay then employ a maximum-likelihood sequence estimation (MLSE) detectorhaving its detection algorithm, such as a Viterbi algorithm (VA),matched to the target response. Thus, state, branch, and path metriccalculations for a Viterbi trellis may be optimized for a channel havingthis target response.

[0013] In accordance with one embodiment of the present invention,compensation is applied to samples received from an optical channel byspectrally shaping, with an equalizer, the samples for compensation togenerate a sequence of equalized samples. An error is generated for acurrent sample based on the difference between 1) an equalized currentsample and 2) a decision for the current sample adjusted for a targetresponse, wherein the target response is based on a response of theoptical channel. The error is combined with one or more samples toprovide an update signal; and the equalizer is updated with the updatesignal to adjust the equalizer response to the target response.

BRIEF DESCRIPTION OF THE DRAWINGS

[0014] Other aspects, features, and advantages of the present inventionwill become more fully apparent from the following detailed description,the appended claims, and the accompanying drawings in which:

[0015]FIG. 1 shows a block diagram of a transmission system including aspectral-shaping detector operating in accordance with an exemplaryembodiment of the present invention;

[0016]FIG. 2 shows an exemplary 2-state transition diagram for the2-state VA algorithm with branch metrics based on the parameter A fortransitions;

[0017]FIG. 3 shows a block diagram of the spectral-shaping detector ofFIG. 2; and

[0018]FIG. 4 shows a block diagram of the finite impulse response (FIR)filter of FIG. 3.

DETAILED DESCRIPTION

[0019] In accordance with exemplary embodiments of the presentinvention, an output signal of a single mode fiber (SMF) is spectrallyshaped to achieve characteristics of a predefined channel (“target”)response. The target response is that of a partial-response channel withadditive white Gaussian noise (AWGN) (e.g., controlled inter-symbolintereference (ISI) is introduced). A receiver employs amaximum-likelihood sequence estimation (MLSE) detector having itsdetection algorithm, such as a Viterbi algorithm (VA), matched to thetarget response. Thus, state, branch, and path metric calculations for aViterbi trellis may be optimized for a channel having this targetresponse. A receiver employing one or more embodiments of the presentinvention may provide improved bit error rate (BER) performance fordetecting and receiving data from a carrier-modulated wavelength from aSMF.

[0020]FIG. 1 shows a block diagram of a transmission system including aspectral-shaping detector 205 operating in accordance with an exemplaryembodiment of the present invention. Data d is transmitted through thechannel comprising SMF 201 to which white, Gaussian noise is added, asshown conceptually as being added to the signal by adder 202. The signalfrom the channel is applied to matched filter 203 to reconstruct thetransmitted pulses including the channel effects. The output of matchedfilter 203 is digitized into input samples with sampling frequency 1/T.The input samples are then applied to spectral-shaping MLSE detector 205operating in accordance with an exemplary embodiment of the presentinvention.

[0021] As described herein, an SMF with data transmission operating at10-Gbps with (τ_(d))≦100 ps is characterized by a partial response withmemory length one. Data generated for differential group delays of 70 psand 100 ps and a given sampling phase may typically be transformed bythe channel into data dominated by ISI of the type a+bD where a=1,b˜0.3. From equation (1), the SMF channel response H(D) has the form ofa+bD, where D is the delay operator (the channel dispersion length istwo symbols).

[0022] Thus, data may be recovered from input samples from an SMFchannel response H(D) of memory length one using a 2-state VA algorithm.The 2-state Viterbi detector is sufficient to de-convolve the ISI foroptimum sequence estimation in the presence of AWGN. SNR loss from ISI(SNR_(loss) ^(ISI)) may be completely recovered by MLSE detection via a2-state VA algorithm given normalized minimum (Euclidean) distanceδ_(min) for a dispersive channel (memory length 1), as in equation (3):

∂_(min) ²=1=>SNR _(loss) ^(ISI)=10log₁₀(∂_(min) ²)=0  (3)

[0023] The present invention is not limited to these data transmissionrates, memory lengths, and SMF channel response forms. In addition, morecomplex forms of target response may be selected.

[0024] For the preferred embodiments, the target response T(D) isselected as A+D, where A is a parameter that for some embodiments may befixed. However, preferred embodiments of the present inventionadaptively select A based on input samples over time, where A may varybetween 0 and 1. The parameter A is incorporated into the branch metricof the two state VA algorithm. FIG. 2 shows an exemplary 2-statetransition diagram for the 2-state VA algorithm with branch metricsbased on the parameter A for transitions. As shown in FIG. 2, the twosymbol states are “1” and “−1”. The branch metric for staying in state“1” is 1/(1+A) and for transitioning to state “−1” is −1/(A−1). Thebranch metric for staying in state “−1” is −1/(−1−A) and fortransitioning to state “1” is 1/(1−A).

[0025]FIG. 3 shows a block diagram of the spectral-shaping, MLSE(SS-MLSE) detector 205 of FIG. 2. SS-MLSE detector 205 comprises SSfinite impulse response (SS-FIR) filter 301, MLSE detector 302,threshold detector 303, response filter 304, combiner 305, multiplier306, and squared-error accumulator 307.

[0026] In a first path, input sample x_(k) at time k is applied toSS-FIR filter 301. As employed herein, the terms x(k) and x_(k) areequivalent notation for the sample at time k. SS-FIR filter 301 appliesequalization to the sample x_(k) to generate the output value y_(k).Equalization of SS-FIR filter 301 adjusts the input samples tocorrespond to samples passing through a channel with the target responseA+D. Filter taps of SS-FIR filter 301 are set based on an update signalu_(k) generated by multiplier 306 in the second path, as describedsubsequently. The output y_(k) of SS-FIR filter 301 at time k may begiven as in equation (4): $\begin{matrix}{{y_{k}\left( c_{k}^{(l)} \right)} = {\sum\limits_{l}{c_{k}^{(l)}{x\left( {{k\quad T} + \tau_{k} - {l\quad T}} \right)}}}} & (4)\end{matrix}$

[0027] where c_(k) ⁽¹⁾ is the l^(th) tap coefficient of FIR filter 301at time k. The equalized output samples y_(k) are then applied to MLSEdetector 302 which detects the transmitted data symbols based on the2-state VA algorithm employing the 2-state transition diagram of FIG. 2.

[0028] Depending on the (converged) sampling phase τ_(k), differentimpulse response coefficients h_(k) characterize the “sampled channel.”Therefore, since the sampled channel shows spectral variation as afunction of sampling phase τ, both equalization and timing recoveryloops may be jointly adapted via a common error signal e. The errorsignal may be derived as known in the art based on a cost function, suchas the mean squared error, and the error signal is derived as thestochastic gradient (or approximation thereof) of the cost function,such as the least means squares (LMS) error. Preferred embodiments ofthe present invention employ a LMS error term for the common errorsignal e.

[0029] Returning to FIG. 3, in a second path input sample x_(k) at timek is applied to threshold detector 303 which generates a decisioncorresponding to the input sample x_(k). During initialization(described subsequently), switch 320 is in position A to provide thedecision of threshold detector 303 to response filter 301. During steadystate operation, switch 320 is in position B to provide the decision ofMLSE detector 302 to response filter 301. Either decision from switch320 is applied to response filter 301 to generate an approximation tothe data passing through a channel with the target response A+D.Combiner 305 generates the error signal (e_(k)) at time k as thedifference between the output y_(k) of (equalizer) SS-FIR filter 301 andthe desired target response signal z_(k) generated by response filter304.

[0030] The common cost function employed is the quadratic error e_(k)⁽¹⁾, which is the highest variance unbiased estimate of the mean-squarederror E(e² _(k)). As shown in Table 1, this cost function is minimizedwith respect to either sampling phase τ_(k) for timing recovery and FIRfilter tap c^((l)) _(k) for equalization. Table 1 also gives therecursive update equation for each cost function. The recursive updateequations are an approximation for updated sampling phase τ_(k) and FIRfilter taps c^((l)) _(k). TABLE 1 Optimization Loop Cost Function e_(k)² Update Equation Sampling Phase$\frac{\partial{e_{k}^{2}\left( {\tau_{k},c_{k}^{(l)}} \right)}}{\partial\tau_{k}} = 0$

τ_(k + 1) = τ_(k) − 2e_(k)(y_(k − 1) − y_(k + 1))

(LMS) Filter Taps (μ is a predetermined constant)$\frac{\partial{e_{k}^{2}\left( {\tau_{k},c_{k}^{(l)}} \right)}}{\partial c_{k}^{(l)}} = 0$

c_(k + 1)^((l)) = c_(k)^((l)) − μe_(k)x_(k)

[0031] The constant μ controls the rate of adaptation, and for preferredembodiments the constant μ may be selected between 0μ<(2/λ_(max)), whereλ_(max) is the largest eigenvalue of the autocorrelation matrix of afilter-tap input vector. The error signal e_(k) is multiplied by theconstant μ and the current sample x_(k) in multiplier 306 to generatethe update signal u_(k).

[0032]FIG. 4 shows a block diagram of an implementation of SS-FIR filter301 of FIG. 3 using a 3-tap adaptive FIR filter. One skilled in the artwould realize that the present invention is not limited to 3 taps, andthat a given implementation of SS-FIR filter 301 may include more orless taps. The 3-tap FIR filter employs LMS adaptation with recursiveupdates as given in Table 1 when operating as an equalizer. SS-FIRfilter 301 comprises delays 401 and 402 generating x_(k−1) and x_(k−2)from x_(k), respectively. Multiplier 306 of FIG. 3 is implemented byfour multipliers 306(a)-306(d). Multiplier 306(d) combines the errore_(k) with the constant μ, and multipliers 306(a)-306(c) multiply x_(k),x_(k−1), and x_(k−2), respectively, with μe_(k).

[0033] Delay 403 and combiner 406 operate to update the FIR filter tapc_(k) ⁽¹⁾ with the output of multiplier 306(a) with recursive update ofTable 1. FIR filter tap c_(k) ⁽²⁾⁻¹ is similarly updated by delay 404and combiner 407, and FIR filter tap c_(k) ⁽³⁾⁻² is similarly updated bydelay 405 and combiner 408. FIR filter tap c_(k) ⁽¹⁾ and current samplex_(k) are multiplied in multiplier 410, FIR filter tap c_(k) ⁽²⁾⁻¹ andprevious sample x_(k−1) are multiplied in multiplier 411, and FIR filtertap c_(k) ⁽³⁾⁻² and sample x_(k−2) are multiplied in multiplier 412. Theoutput values of multipliers 410, 411, and 412 are added together incombiners 413 and 414 to generate the current filtered sample y_(k).

[0034] Since the sampled channel is spectrally close to the targetresponse, the equalizer's noise enhancement and tap length N may bereduced. For the preferred embodiment, the precision (tap width) forfilter tap values is 15 bits. The LMS adaptation takes place for everysymbol (full baud rate) the rate for such recursive adaptation may bereduced by performing adaptation for sequential blocks of symbols (e.g.every 20 symbols). The adaptation rate may depend on the rate of channelvariation.

[0035] Returning to FIG. 3, during start-up or other initialization, theparameter A may be initially set with an adaptation process as follows.Squared-error accumulator 307 generates the current mean squared error(MSE) by accumulating e_(k) ² for each k. The current MSE is a quantityrelated to the channel quality. Parameter A may be defined with adiscrete set of levels, such as the set A={0.2, 0.4, 0.6, 0.8, 1}. Thecurrent MSE value may then be employed to select a value for A from thediscrete set of levels. The parameter A is adapted using an adjustmentstep of, for example, 0.2. Thus, the values for tap coefficients of FIRfilter 301 are set to an initial value. Then the value of current MSEmay be compared to an adaptation threshold, and while the value ofcurrent MSE is greater than the adaptation threshold, the parameter A isincremented by the adaptation step (when starting from a low initialvalue for A). When the current MSE is less than the adaptationthreshold, the value for A is fixed, and taps of FIR filter 301 are thenadapted. Alternatively, f starting from a high initial value for A, theadaptation process may decrement by steps until the current MSE is lessthan the adaptation threshold. During initialization, the branch metricsof the 2-state VA such as shown in FIG. 2 may also be updated, or may befixed until the end of the initialization process, at which time thevalue of A for the 2-state VA is fixed.

[0036] During operation, some embodiments may also adaptively update thevalue of A during steady-state operation by monitoring the value of thecurrent MSE. For this case, if the value of MSE becomes greater than theadaptation threshold, then the parameter A is incremented by theadaptation step until the current MSE is less than the adaptationthreshold. To reduce A, the parameter A is periodically decremented byan adaptation step and the current MSE compared with the adaptationthreshold. If the current MSE is greater than the adaptation threshold,the value for A is incremented by the adaptation step, but, if thecurrent MSE remains less than the adaptation threshold, the value for Ais decremented again by the adaptation step until the current MSE isgreater than the adaptation threshold, at which point the parameter forA is incremented by the adaptation step and set. Again, the branchmetrics of the 2-state VA such as shown in FIG. 2 may also be updatedduring this process.

[0037] While the exemplary embodiments of the present invention havebeen described with respect to processes of circuits, the presentinvention is not so limited. As would be apparent to one skilled in theart, various functions of circuit elements may also be implemented inthe digital domain as processing steps in a software program. Suchsoftware may be employed in, for example, a digital signal processor,microcontroller or general purpose computer.

[0038] The present invention can be embodied in the form of methods andapparatuses for practicing those methods. The present invention can alsobe embodied in the form of program code embodied in tangible media, suchas floppy diskettes, CD-ROMs, hard drives, or any other machine-readablestorage medium, wherein, when the program code is loaded into andexecuted by a machine, such as a computer, the machine becomes anapparatus for practicing the invention. The present invention can alsobe embodied in the form of program code, for example, whether stored ina storage medium, loaded into and/or executed by a machine, ortransmitted over some transmission medium, such as over electricalwiring or cabling, through fiber optics, or via electromagneticradiation, wherein, when the program code is loaded into and executed bya machine, such as a computer, the machine becomes an apparatus forpracticing the invention. When implemented on a general-purposeprocessor, the program code segments combine with the processor toprovide a unique device that operates analogously to specific logiccircuits.

[0039] It will be further understood that various changes in thedetails, materials, and arrangements of the parts which have beendescribed and illustrated in order to explain the nature of thisinvention may be made by those skilled in the art without departing fromthe principle and scope of the invention as expressed in the followingclaims.

What is claimed is:
 1. An apparatus for applying compensation to samplesreceived from an optical channel comprising: an equalizer having anequalizer response spectrally shaping the samples for compensation togenerate a sequence of equalized samples; an error generator generatingan error for a current sample based on the difference between 1) anequalized current sample and 2) a decision for the current sampleadjusted for a target response, wherein the target response is based ona response of the optical channel; and a combiner configured to combinethe error with one or more samples to provide an update signal, whereinthe equalizer employs the update signal to adjust the equalizer responseto the target response.
 2. The invention as recited in claim 1, furthercomprising a maximum likelihood sequence estimation (MLSE) detector, theMLSE detector generating decoded data from the sequence of equalizedsamples.
 3. The invention as recited in claim 2, wherein the MLSEdetector generates decoded data with an algorithm having transitionsbased on the target response.
 4. The invention as recited in claim 3,further comprising an accumulator configured to accumulate the square ofeach error value, wherein the accumulation of the squared error valuesrelates to a parameter of the target response, and the algorithm adjustsits transitions by adaptation of the parameter of the target response.5. The invention as recited in claim 1, wherein the equalizer comprisesa filter defined by a set of filter taps.
 6. The invention as recited inclaim 5, wherein the set of filter taps are adapted in accordance with arecursive update rule, wherein the update rule is generated from a costfunction.
 7. The invention as recited in claim 5, wherein the costfunction is quadratic error and the update rule is generated fromminimizing mean squared error of the cost function with respect to thefilter tap.
 8. The invention as recited in claim 5, wherein the targetresponse is of the form A+D, where A is a parameter ranging from about 0to about 1, and D is a unit delay.
 9. The invention as recited in claim1, further comprising an accumulator configured to accumulate the squareof each error value, wherein the accumulation of the squared errorvalues relates to a parameter of the target response, and the apparatusadapts the parameter of the target response during initialization of theapparatus.
 10. The invention as recited in claim 1, wherein theequalization applied to the current sample accounts for differentialgroup delay of a signal passing through a single mode fiber.
 11. Theinvention as recited in claim 1, wherein the apparatus is embodied in anintegrated circuit.
 12. The invention as recited in claim 1, wherein theapparatus is implemented in a receiver of an optical communicationterminal.
 13. A method of applying compensation to samples received froman optical channel comprising the steps of: (a) spectrally shaping, withan equalizer, the samples for compensation to generate a sequence ofequalized samples; (b) generating an error for a current sample based onthe difference between 1) an equalized current sample and 2) a decisionfor the current sample adjusted for a target response, wherein thetarget response is based on a response of the optical channel; (c)combining the error with one or more samples to provide an updatesignal; and (d) updating the equalizer with the update signal to adjustthe equalizer response to the target response.
 14. The invention asrecited in claim 13, further comprising the step of (e) generatingdecoded data from the sequence of equalized samples with maximumlikelihood sequence estimation (MLSE) detection.
 15. The invention asrecited in claim 14, wherein step (e) generates decoded data with analgorithm having transitions based on the target response.
 16. Theinvention as recited in claim 15, further comprising the steps ofaccumulating the square of each error value, wherein the accumulation ofthe squared error values relates to a parameter of the target response,and adjusting its transitions by adaptation of the parameter of thetarget response.
 17. The invention as recited in claim 13, wherein step(a) comprises the step (a1) of filtering based on a set of filter taps.18. The invention as recited in claim 17, wherein step (a1) includes thestep of adapting the set of filter taps in accordance with a recursiveupdate rule based on a cost function.
 19. The invention as recited inclaim 18, wherein the cost function is quadratic error and the updaterule is generated from minimizing mean squared error of the costfunction with respect to the filter tap.
 20. The invention as recited inclaim 17, wherein, for step (b) the target response is of the form A+D,where A is a parameter ranging from about 0 to about 1, and D is a unitdelay.
 21. The invention as recited in claim 13, further comprising thesteps of: accumulating the square of each error value, wherein theaccumulation of the squared error values relates to a parameter of thetarget response, and adapting the parameter of the target responseduring initialization of the apparatus.
 22. The invention as recited inclaim 13, wherein for step (a), the compensation applied to the currentsample accounts for differential group delay of a signal passing througha single mode fiber.
 23. The invention as recited in claim 13, whereinthe method is embodied in a processor of an integrated circuit.
 24. Theinvention as recited in claim 13, wherein the method is embodied in areceiver of an optical communication terminal.
 25. A computer-readablemedium having stored thereon a plurality of instructions, the pluralityof instructions including instructions which, when executed by aprocessor, cause the processor to implement a method for applyingcompensation to samples received from an optical channel, the methodcomprising the steps of: (a) spectrally shaping, with an equalizer, thesamples to generate a sequence of equalized samples; (b) generating anerror for a current sample based on the difference between 1) anequalized current sample and 2) a decision for the current sampleadjusted for a target response, wherein the target response is based ona response of the optical channel; (c) combining the error with one ormore samples to provide an update signal for each tap of the equalizer;and (d) updating the equalizer with the update signal to adjust theequalizer response to the target response.